On relations between one-dimensional quantum and two-dimensional classical spin systems

TL;DR

This paper explores the deep connections between one-dimensional quantum and two-dimensional classical spin systems, establishing universality of their critical phenomena through multiple mapping techniques.

Contribution

It introduces new methods to relate quantum and classical systems, confirming the universality of critical behavior across these models.

Findings

Critical properties are equivalent in mapped quantum and classical systems.

Multiple approaches confirm the universality of critical phenomena.

The work extends previous results on classical systems to quantum counterparts.

Abstract

We exploit mappings between quantum and classical systems in order to obtain a class of two-dimensional classical systems with critical properties equivalent to those of the class of one-dimensional quantum systems discussed in a companion paper (J. Hutchinson, J. P. Keating, and F. Mezzadri, arXiv:1503.05732). In particular, we use three approaches: the Trotter-Suzuki mapping; the method of coherent states; and a calculation based on commuting the quantum Hamiltonian with the transfer matrix of a classical system. This enables us to establish universality of certain critical phenomena by extension from the results in our previous article for the classical systems identified.